Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5,26,Mod(1,5)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(19.7998389976\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 31823342x^{3} + 3040467992x^{2} + 155755658754016x - 41401144140044416 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{3}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{5} - x^{4} - 31823342x^{3} + 3040467992x^{2} + 155755658754016x - 41401144140044416 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{4} + 5339\nu^{3} + 204537534\nu^{2} - 118916988560\nu - 630583703059808 ) / 446010624 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 49\nu^{4} - 37373\nu^{3} + 352279758\nu^{2} + 1083968911856\nu - 18295641739665760 ) / 446010624 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7333\nu^{4} - 22725151\nu^{3} + 211067727450\nu^{2} + 499281649962832\nu - 671092626032517152 ) / 446010624 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 7\beta_{2} - 282\beta _1 + 50917459 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -63\beta_{4} - 113\beta_{3} + 65206\beta_{2} + 44092373\beta _1 - 7238397117 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -48051\beta_{4} + 29133461\beta_{3} - 592132\beta_{2} - 8586340323\beta _1 + 1121935840994173 ) / 4 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−11003.9 | −418448. | 8.75320e7 | 2.44141e8 | 4.60457e9 | 7.06271e10 | −5.93966e11 | −6.72190e11 | −2.68651e12 | |||||||||||||||||||||||||||||||||
| 1.2 | −5620.55 | −161492. | −1.96387e6 | 2.44141e8 | 9.07675e8 | −4.86926e10 | 1.99632e11 | −8.21209e11 | −1.37220e12 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1456.68 | 1.56404e6 | −3.14325e7 | 2.44141e8 | −2.27831e9 | 3.49059e10 | 9.46654e10 | 1.59893e12 | −3.55636e11 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4172.52 | −1.12842e6 | −1.61445e7 | 2.44141e8 | −4.70836e9 | −2.36507e10 | −2.07370e11 | 4.26045e11 | 1.01868e12 | ||||||||||||||||||||||||||||||||||
| 1.5 | 9306.64 | 770525. | 5.30591e7 | 2.44141e8 | 7.17100e9 | 2.22916e10 | 1.81523e11 | −2.53579e11 | 2.27213e12 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5.26.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 45.26.a.f | 5 | ||
| 5.b | even | 2 | 1 | 25.26.a.c | 5 | ||
| 5.c | odd | 4 | 2 | 25.26.b.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.26.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 25.26.a.c | 5 | 5.b | even | 2 | 1 | ||
| 25.26.b.c | 10 | 5.c | odd | 4 | 2 | ||
| 45.26.a.f | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 4602T_{2}^{4} - 118822008T_{2}^{3} - 367856402816T_{2}^{2} + 2127699725500416T_{2} + 3498510988641042432 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 34\!\cdots\!32 \)
|
| $3$ |
\( T^{5} + \cdots + 91\!\cdots\!76 \)
|
| $5$ |
\( (T - 244140625)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 63\!\cdots\!68 \)
|
| $11$ |
\( T^{5} + \cdots + 87\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots - 56\!\cdots\!24 \)
|
| $17$ |
\( T^{5} + \cdots + 11\!\cdots\!32 \)
|
| $19$ |
\( T^{5} + \cdots - 68\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 70\!\cdots\!76 \)
|
| $29$ |
\( T^{5} + \cdots - 52\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 12\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots - 97\!\cdots\!68 \)
|
| $41$ |
\( T^{5} + \cdots - 29\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots + 65\!\cdots\!76 \)
|
| $47$ |
\( T^{5} + \cdots + 44\!\cdots\!32 \)
|
| $53$ |
\( T^{5} + \cdots + 15\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots + 84\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 53\!\cdots\!68 \)
|
| $67$ |
\( T^{5} + \cdots - 16\!\cdots\!68 \)
|
| $71$ |
\( T^{5} + \cdots + 32\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 19\!\cdots\!24 \)
|
| $79$ |
\( T^{5} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 21\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 47\!\cdots\!68 \)
|
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